ANNALES KINESIOLOGIAE • 1 • 2010 • 1 29 A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY TO PHILOSOPHY Pietro Enrico DI PRAMPERO 1 1 University of Udine, Department of Biomedical Sciences, P:le M. Kolbe 4, 33100 Udine, Italy e-mail: pietro.prampero@uniud.it ABSTRACT The development of a scientifi c theory (T) can be separated into successive phases: i) Fantasy, to conceive T ii) Analysis to couch T into formal language iii) Action, to apply in practice the predictions of T. The history of human powered fl ight, in which case the three phases are stretched over several thousand years, allow us to better appreciate their intrinsic characteristics. Fantasy, dating back to the myth of Ikarus, must be experimentally testable, as indeed were Daedalus’ wings. Analysis must state in quantitative terms the laws governing the matter at stake. Action, from Leonardo’s unsuccessful attempts to the crossings of the British Channel in 1979 and of the arm of the sea separating Crete from mainland Greece in 1988, has the aim of shaping the world according to our will. The kernel of any “proper” T is a formal system wherein a set of operational rules allows us to manipulate a set of symbols, representing the objects of T, on the bases of a limited number of axioms. In such formal systems, “theo- rem” is a string of symbols that can be arrived at in a fi nite number of steps from the axioms, applying the canonical operational rules. However, as Kurt Gödel showed in 1931, it is possible to demonstrate that, within a suffi ciently powerful formal system, there exists demonstrably true strings of symbols that are not theorems. Thus, even in an ultra-powerful theory of everything, there will still be truths that can not be arrived at within the theory. Keywords: scientifi c method, fl ight by birds, human-powered fl ight, maximal aero- bic power, body mass, scientifi c truth original scientifi c article UDC: 797.5:001.1 received: 2010-06-08 ANNALES KINESIOLOGIAE • 1 • 2010 • 1 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ...,29–45 30 KRATKA ZGODOVINA LETENJA NA ČLOVEŠKI POGON: OD FIZIOLOGIJE DO FILOZOFIJE IZVLEČEK Razvoj znanstvene teorije (T) se lahko razdeli na naslednje faze: i) fantazijo, ki zasnuje T, ii) analizo, v kateri se T ubesedi v formalnem jeziku, in iii) akcijo, kjer se predvidevanja T uporabi v praksi. Zgodovina letenja na človeški pogon, v kateri se te tri faze razpotegnejo skozi nekaj tisočletij, nam omogoča bolje razumeti njihove značilnosti. Fantazijo, ki jo lahko datiramo vse do mita o Ikarusu, se mora dati eksperi- mentalno preizkusiti, kar se je v tem primeru dalo z Dedalovimi krili. Analiza mora v količinskem pogledu opisati zakone, ki opredeljujejo zamišljeno zadevo. Akcija, od Leonardovih neuspešnih poskusov do prečkanja Rokavskega preliva leta 1979, 1875 in morja, ki loči Kreto od celinske Grčije leta 1988, ima za cilj oblikovati svet po naši volji. Jedro vsake “prave”T je formalni sistem, v katerem nam nabor operativnih pravil omogoča operirati z naborom simbolov, ki predstavljajo predmete T, na podlagi omejenega števila aksiomov. V takih formalnih sistemih je “teorem” niz simbolov, do katerega se pride s končnim številom korakov iz aksiomov z uporabo kanonskih op- erativnih pravil. Vendar, kot je leta 1931 pokazal Kurt Gödel, je možno dokazati, da obstajajo v dovolj močnem formalnem sistemu dokazljivo resnični nabori simbolov, ki niso teoremi. Torej bodo tudi v zelo mogočni teoriji vsega še vedno resnice, do katerih ni mogoče priti znotraj teorije. Ključne besede: znanstvena metoda, letenje ptic, letenje na človeški pogon, največ- ja aerobna moč, masa telesa, znanstvena resnica INTRODUCTION The interest in the philosophy of science has gained substantial momentum not only among philosophers, but also among scientists and cultivated laymen. The reasons for this will not be discussed here; however, it is likely due (at least in part) to both the spectacular success of applied science and technology and to the fears that this success inevitably brings. Be this as it may, it seemed to me that the viewpoint on this matter of a professional physiologist and “naif philosopher” may be of some interest to the reader. So, the aim of what follows is a brief outline of the structure of science. After having stated the basic tenets of the Galilean scientifi c method, I will attempt to demon- strate that the development of a scientifi c idea can be separated into successive, partly overlapping phases: i) Fantasy, to conceive a scientifi c theory ii) Analysis, to couch the Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ..., 29–45 ANNALES KINESIOLOGIAE • 1 • 2010 • 1 31 theory into formal language iii) Action, to apply in practice the prediction of the theory. To this aim, I will briefl y summarise the history of human powered fl ight, in which case the more than two millennia over which the three phases are stretched allows us to bet- ter appreciate them. Many historians and philosophers of science have pointed out, however, that the Galilean method, wherein the agreement between facts and theory plays a pivotal role in the acceptance of a given theory, is a somewhat simplifi ed idealisation. Indeed, sev- eral other characteristics foster the acceptance of a theory in addition to the agreement between predictions and facts. These points of view can be reconciled with the more traditional Galilean science by means of “The Catastrophe Theory”, as described by René Thom, which allows us to view the acceptability of a given theory within the scientifi c community as being dependent not only on the agreement between predic- tions and facts, but also on several (indeed as many as we can possibly conceive) other factors. In the concluding paragraphs I will address the meaning of “scientifi c truth” and I will try to convince the reader that even assuming that in the very distant future science will be condensed in a “ultra-powerful” Theory of Everything, as shown by Kurt Gödel in 1931 for suffi ciently powerful formal systems, there will still be Truths that can not be arrived at within the theory. THE GALILEAN SCIENTIFIC METHOD The Galilean scientifi c method can be schematically represented as in Figure 1. Here the predictions derived from a given theory are compared to the results obtained from appropriated experiments and/or from observations of the external world. This process is assumed to yield a continuous refi nement of the theory at stake, thus gradu- ally approaching “scientifi c truth”. The process summarised in Figure 1 has probably been, albeit implicitly, at the root of human knowledge since the dawn of mankind. However, it has been made explicit, at least in western culture essentially by the works and writings of Galileo Galilei (1564– 1642), and has since become the main pillar of the scientifi c method. As shown in Figure 1, the Galilean method can be loosely, if somewhat artifi cially, separated into three phases. In the fi rst phase, which will be defi ned here as “Fantasy”, scientifi c theories are forged. This is followed by a second one “Analysis” in which sci- entifi c theories are couched into quantitative formal language, necessary for extracting predictions from theories and for planning experiments and/or observations allowing the investigator to compare facts and theories. Finally, in the last phase “Action”, theoreti- cal knowledge is utilised in practice to shape the external world according to our will. Furthermore, as briefl y discussed below, the main intellectual and practical tools utilised in the three phases are widely different; a few examples are indicated in Figure 1. ANNALES KINESIOLOGIAE • 1 • 2010 • 1 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ...,29–45 32 As a general rule, in the scientifi c routine, these phases largely overlap and rever- berate onto each other; as such they can not be clearly separated. However, it seems to me that upon close scrutiny, they can be easily identifi ed in the conceptual development of many, if not all, scientifi c processes. A particularly fruitful example is the historical evolution of human powered fl ight, wherein the three phases “Fantasy”, “Analysis”, “Action”, spread as they are over millennia, can be easily identifi ed. The sections that follow will therefore be dedicated to a brief recapitulation of the history of human powered fl ight. Figure 1: The scientifi c method. The predictions deduced from a given theory are com- pared with the results of appropriate experiments and/or observations and/or manipu- lations of the external world. This is assumed to yield a gradual refi nement of the theory. The process is subdivided into three phases. The main intellectual and practical tools utilised in the three phases are also indicated. See text for details. Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ..., 29–45 ANNALES KINESIOLOGIAE • 1 • 2010 • 1 33 Fantasy Postquam manus ultimas coeptis imposita est, geminas opifex libravit in alas ipse suum corpus, motaque pependit in aura.1 Daedalus was the famous Greek architect who built the Labyrinth where Minos, King of Crete, sequestered his son, the Minotaur, who had the body of a man and the head of a bull. To avenge the death of another son, Androgeus killed by the Athenians, Minos demanded that seven Athenian youth and seven maidens should be sent every ninth year to be devoured by the Minotaur. Theseus volunteered to be among the nine youths to be sent to Crete, where he entered the Labyrinth and killed the Minotaur, thus freeing the Athenians form the horrible tribute. Theseus succeeded in fi nding the way out of the Labyrinth thanks to Ariadne who fell in love with him a gave him a spool of thread to guide him through the mazes of the Labyrinth. In turn, this trick had been suggested to Ariadne by Daedalus. Minos discovered the whole plot and sequestered Daedalus and his son Icarus in the Labyrinth. In spite of being the architect of the Labyrinth, Daedalus was not able to fi nd the way out. If not by land, then by air: Daedalus constructed four large wings with feathers and wax and glued them to Icarus’ and his shoulders. They are up, in the air. Thrilled by success, Icarus fl ies high, too close to the Sun, the heat of which melts the wax. Icarus falls into the Egean Sea, whereas a tearful Daedalus, reaches Sicily where King Cocalus welcomes and protects him. So far, the myth is the obvious product of fantasy. This, however, is somehow re- alistic: the wings constructed by Daedalus are based on a model provided by Mother Nature; as such the model can be experimentally tested. More than two thousand years will have to elapse before formal proof could be reached that to fl y using his own muscle power, man should abandon the idea of imitating Nature. Even so, Daedalus’ fantasy is the kernel of a scientifi c enterprise which will eventually succeed. Thus, the fundamental requirement of scientifi c Fantasy, in the sense utilised in this article, is that the products thereof be experimentally testable. As such, scientifi c Fantasy is conceptu- ally different from artistic or poetic fantasy which does not have any requirements in terms of experimental verifi cation. 1 “After having given the last touch to his work, the craftsman lifted his own body on twin wings, and suspended it into the moving air.” Ovid (Publius Ovidius Naso 43B.C.–18A.D.), Metamorphoses VIII: 200–202. ANNALES KINESIOLOGIAE • 1 • 2010 • 1 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ...,29–45 34 Analysis Sicuti terrestria animalia super terram, sic aves per aere volando incedunt. Talis motus effi citur mirabili artifi cio, et organis mechanicis, quorum theoriam esplicare conabimur.2 The fi rst descriptions of the macroscopic anatomy of birds are due to Aristotle (384-322 B.C.) and to his disciples. After very nearly two millennia without substan- tial innovations, Leonardo da Vinci (1452-1519) dedicates a non negligible fraction of his monumental opus to study animal and human fl ight. Leonardo is not content with knowledge, he strives for transferring it to practice: to have man fl y like a bird. His drawings show his genius, but his practical attempts met with failure. One hundred and fi fty years after Leonardo’s death, the physiologist Giovanni Al- fonso Borelli (1608-1679) devotes a fraction of his book “De Motu Animalium” (which can be considered the forerunner of the modern textbooks on biomechanics) to the fl ight of birds and humans. Galileo had previously shown that the mass of geometrical- ly similar animals increases with the cube of a given linear dimension (since their body density is essentially equal), whereas their surface increases with the square of the same linear dimension. Extending this type of analysis to birds, Borelli shows that a given surface of the wing of a bird must sustain a weight which increases with the dimensions of the animal. From this line of reasoning, Borelli draws two conclusions: a) there must exist an upper limit of body mass above which no bird, conceived according to the rules of Mother Nature, can possibly fl y, and b) the muscles of the upper girdle and limbs of humans are too weak to allow them to fl y like birds. Conclusion b shows why Daedalus’ idea, if applied literally, was doomed to failure. Indeed, to fl y with his own muscular power, man has followed a different route, which will be described below. About three hundred and fi fty years after Borelli, his ideas were given a quantita- tive from, thanks to the work of the British physiologist Douglas Wilkie (1922-1998). Wilkie showed that, for a given set of geometrically similar birds of equal body density, the metabolic power requirement to fl y horizontally at the minimum (constant) speed necessary for sustaining the animal in the air (Pf), increases with the mass (M) of the animal as described by: 2 “As the beasts of the earth move on the land, so do the Birds wing their fl ight through the air. This mo- tion of fl ight is accomplished with marvellous skill and by means of organic mechanism in such fashion as we shall here endeavour to set out”. G. Alfonso Borelli. De Motu Animalium -Caput XXII. Romae ex Typographia Angeli Bernabò, MDCLXXX. English translation by T.O’B. Habbard, J.H. Ledeboer, in “Founders of Experimental Physiology - Biographies and Translations”. Ed. by John W. Boylan. J.F. Lehmanns Verlag, Munich, 1971, p. 23 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ..., 29–45 ANNALES KINESIOLOGIAE • 1 • 2010 • 1 35 Pf = ß M 1.167 1) where ß is a constant which depends on the shape of the animal, on the air den- sity, the acceleration of gravity and the effi ciency of transformation of metabolic into mechanical power (Wilkie, 1959). It is worth noting that Equation 1 applies also to geometrical similar airplanes, in which case Pf represents the power of the engine and ß assumes, obviously enough, a different value. Equation 1 shows that the minimum metabolic power for constant speed horizontal fl ight at the lowest possible speed in- creases more than the mass of the animal: e.g., for an increase of M of four times, Pf increases fi ve times. At variance with Pf, the metabolic power that the animals can actually generate (E) increases less than the mass of the animal: E = a M 0.79 2) where a is a constant which depends on the duration of the exercise (Taylor et al., 1980). For E in watts and M in kg, a is about 4 for resting metabolism, 24 for the meta- bolic power sustainable over a 6 hour period, and 40 for the maximal aerobic power, which can be sustained for about 10 minutes. Equation 2 shows that the increase of E associated with a fourfold increase of mass is only threefold, as compared to the fi ve- fold increase of Pf mentioned above. Equations 1 and 2 taken together show the conundrum in which Mother Nature has put herself when producing fl ying animals of greater and greater body mass: with increasing body mass, the minimum power required for fl ight increases more that the power generated by the animal. In quantitative terms, these two equations allow us to calculate the maximal body mass for which horizontal fl ight is a physiological pos- sibility, as a function of the two constants ß and a. Indeed, for fl ight to be possible the minimum metabolic power required must be equal to that generated by the animal (Pf = E). Hence, from equations 1 and 2: ß M 1.167 = a M 0.79 3) or, taking the logarithms: log ß + 1.167 log M = log a + 0.79 log M 3’) and rearranging: log M = (log a - log ß)/0.38 4) ANNALES KINESIOLOGIAE • 1 • 2010 • 1 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ...,29–45 36 For natural fl ight, ß amounts to about 8.3 (for Pf in watts and M in kg). Thus, insert- ing into Equation 4 the two values of a applying for metabolic power sustainable for 6 hours and 10 minutes, M turns out to be 14 and 65 kg, respectively. Thanks to this line of thinking, essentially due to Wilkie, it is therefore possible to assign numerical values to the qualitative predictions of Borelli. Indeed, the maximal possible body mass for a bird capable of sustained fl ight can not exceed 14 kg. Inci- dentally, this is close to the mass of the great ottard, a renowned long distance fl yer (Figure 2). This also shows that migrating birds, the body mass of which is much less than the critical mass of 15 kg, can overcome the enormous distances required by their ecological needs without increasing their metabolic power to a great extent. Indeed, rearranging equation 3’: log a = log ß + log M (1.167 - 0.79) Since ß = 8.3, assuming e.g. M = 0.1 kg, a turns out to be 3.5, i.e. close to that for resting metabolism. This value applies to the minimal speed compatible with horizontal fl ight, thus showing that the migrating bird does not need to raise his metabolism to a great extent to achieve the desired migration speed. Figure 2: The great ottard (Otis Tarda). Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ..., 29–45 ANNALES KINESIOLOGIAE • 1 • 2010 • 1 37 The body mass calculated above for a duration of 10 minutes is close to that of a small size human adult. This shows that, even if humans were constructed like a stand- ard bird, their fl ight capabilities would be essentially nil, since for any realistic situation they should rely on anaerobic metabolism, a fact that would greatly reduce the actual fl ight duration, not to mention take off and turns, in which case, even at the minimal speed the power requirement increases substantially. Once again, these considerations support and extend quantitatively the original qualitative predictions of Borelli. Action Because of Icarus and what befell him the idea of manpowered fl ight started off with bad press. What was really no more than a minor structural failure has been wrongly taken as proof that the whole thing was impossible in principle (Wilkie, 1982). The fi rst attempts to produce fl ight machines moved by human muscle power are due to Leonardo da Vinci. In spite of his ingenuity as witnessed by his magnifi cent drawings, his attempts met with failure. Leaving aside the father of gliding fl ight, Otto Lilienthal (1848-1896) who made over 2000 glides in safety and was killed on August 9, 1896 when his glider was caught by a sudden gust of wind, the fi rst successful man- powered fl ights were accomplished in 1935 - 36 by a German (Hassler-Villiger) and an Italian (Bossi-Bonomi) machine which fl ew 700 and 920 metres. The power gener- ated by the engine and pilot was not enough for take-off which was made possible by a catapult. World War II put a tragic end to these attempts of peaceful aviation which were resumed in the 1950s, thanks to the generosity of an Englishman, Mr. H. Kremer, who established a 5000 pound prize for the fi rst man-powered vehicle to fl y a fi gure eight around two pitons half a mile (800 metres) apart. In spite of numerous attempts by English and Japanese engineers and scientists who achieved straight fl ights over 1000 metres, wherein also take-off was generated by the engine-pilot, the fi gure eight fl ight remained off-limits because of the increased power required for the two turns, and the prize climbed to 10,000 and then 50,000 pounds. The situation changed in the mid 1970s when an America team lead by Paul Mac- Cready constructed a machine characterised by an “old fashion” design similar to that allowing the Wright brothers to accomplish the fi rst mechanical powered fl ight in De- cember 1903. The characteristics of this machine (The Gossamer Condor) was that of generating a lift suffi cient to maintain it and the engine-pilot in the air, already at a very low speed (about 12 km/h), thus reducing to a minimum the power necessary to overcome the air drag which increases with the cube of the speed. This, coupled with an extremely light weight (32 kg) and a very large wing surface area (97.2 m2) allowed ANNALES KINESIOLOGIAE • 1 • 2010 • 1 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ...,29–45 38 the team to win the Kremer fi gure eight prize. On June 12, 1979, a similar machine (The Gossamer Albatross, Figure 3) powered and steered by Brian Allen crossed the English channel from Folkestone to Cap Gis-Nez, near Calais (35.6 km), at an average speed of 12.6 km/h, in spite of relatively strong opposing winds, thus winning a new prize of 100,000 pounds that Mr. Kremer had established in the meantime for the fi rst man-powered machine to achieve this deed (Grosser, 1991). The ideas of Paul MacCready and his team obtained additional recognition on April 23, 1988, when the Greek cyclist K. Kanellopoulos succeeded in fl ying over the arm of sea between Crete and the island of Santorini (120 km) at an average speed of 30.3 km/h, thus showing that Daedalus was not so wrong after all3. 3 For a more detailed discussion of these matters see Prampero (di) P.E. “Il volo a propulsione umana” in “Dagli abissi allo spazio. Ambienti e Limiti umani” Ed. by G. Ferretti, C. Capelli, Edi Ermes, Milano 2008, pp. 151–159. Figure 3: Brian Allen fl ying over the British Channel on June 12, 1979. Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ..., 29–45 ANNALES KINESIOLOGIAE • 1 • 2010 • 1 39 THE SCIENTIFIC METHOD Fatti non foste a viver come bruti, ma per seguir virtude e conoscenza.4 In the previous paragraphs, I have attempted to show that the development of any given scientifi c idea can be subdivided into three phases: “Fantasy” to conceive a the- ory (T), “Analysis” to express T in formal language; “Action” to utilise T in practice. Each phase would deserve a far deeper discussion than was possible in this article. Nevertheless, I would like to point out briefl y the main characteristics of each phase. Fantasy. From where a scientist or a group of scientists begets a theory (T) is a matter of fascinating debate: fortuitous coincidences, chance, analogies, dreams (as in the famous case of the chemist F.A. Kekule von Stradonitz (1829–1896) who allegedly conceived in a dream the cyclic nature of the aromatic compounds of carbon, such as benzene (Holton, 1973, 1978, 1979). Be this as it may, the so conceived T must be somehow realistic. Indeed, to be of any scientifi c value, its products must be experi- mentally testable even if often, at least initially, only by means of the so called “Gedank Experimente” (thought experiments). As mentioned earlier, this is the main difference between scientifi c and artistic fantasy, this last being free from any experimental con- straint. However, common to both is the aesthetic appeal: scientists and artists alike are inevitably partial towards “beautiful” theories. Foremost among the canons of scientifi c beauty is simplicity, as synthesised by Occam’s Razor “Entia non sunt moltiplicanda praeter necessitatem”5. Analysis. This is the phase in which T is expressed in formal, often mathematical terms. It is a crucial step which allows the scientist to derive from T a fi nite set of test- able predictions that can be verifi ed or disproven by experiment and/or observation, the results of which can be utilised to gradually refi ne T. Another fundamental role of this phase is that of expressing T in “public language”, thus making it as well as its assump- tions, predictions, constraints, limits, available in principle to the scientifi c community. Public language is one of the crucial differences between science and other types of knowledge such as religious or mystical beliefs which are free from the stringent neces- sity of expressing their Ts in formal language. On a more mundane note it seems appro- priate to mention here Lord Rutherford’s (1871–1937, Nobel Prize 1908 for Chemistry) famous dictum “Truth comes easier from error than from confusion”, wherein he very effectively stressed the fact that an erroneous scientifi c theory expressed in formal lan- guage is more useful in scientifi c terms than a vast collection of unorganised data. In- 4 “you were not made to live like brutes, but to follow virtue and knowledge.” Dante Alighieri - La Divina Commedia - Inferno, canto XXVI 119–120. 5 William of Occam (c. 1280–1349). “Beings ought not to be multiplied except out of necessity.” ANNALES KINESIOLOGIAE • 1 • 2010 • 1 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ...,29–45 40 deed, in the former case, the theory can be proven wrong and thus be a stimulus towards a new one, whereas a vast array of data or observations without a possible theoretical common ground is less likely to lead to organised science. Action. This is the phase in which the conceptual products of a scientifi c theory are utilised in practice to shape the external world. It is the phase to which the great majority of people will refer to when asked about the meaning of the term science: the phase worshipped for its triumphs or blamed for its feared dark sides. Both extremes, however, miss the beauty of this most human endeavour which encompasses the richest fruits of Western thought: 1) The Greek quest for theorising and knowledge (Fantasy); 2) The Roman drive towards organisation and law (Analysis); 3) The desire to mould the world according to our needs, gaining ever growing momentum during the Renais- sance and Modern times (Action). The process described above and summarised in Figure 1 may suggest that with time and effort scientifi c theories gradually progress towards a state of perfection as- ymptotically approximating Truth. Upon closer refl ection, however, this is far from evi- dent. Indeed, as described so far, the process does not explain what happens when, as is often the case, a given T is not confi rmed by experiments and/or observation. Thus, the simplifi ed view of the process summarised in fi gure 1 may well apply under conditions of “normal science” but fails to do so in the troublesome periods of “scientifi c revolu- tions”. These considerations were brought to the foreground of the debate on the history and philosophy of science be Thomas Kuhn in his remarkable book “The Structure of Scientifi c Revolutions” (Kuhn, 1970). More recently other authors have taken the ex- treme view that the correspondence between experiments and observation, on the one side, and predictions of the theory on the other, plays only a minor or negligible role in the acceptance of a given T by the scientifi c community. According to these authors, the crucial factor in the acceptance of a given theory must be looked for in other aspects of science, such as social conventions or fashion to the extreme position of Feyerabend “anything goes” (Feyerabend, 1975). The majority of active scientists will probably refute these positions, since they know very well how important the agreement between fact and theory is. However, they are also very well aware of the relevance of other factors, such as scientifi c fashion, political interests, the availability of research fund- ing, etc. in directing the main research lines of the scientifi c community (Feyerabend, 1981). The extreme example of this state of affairs is the (un)famous scientifi c disaster of Lysenko’s genetics in URSS under Stalin’s dictatorship. In an attempt to reconcile the role of hard facts (i.e. the agreement between predic- tions and experimental observations) with that of other factors, social, political, aes- thetic, etc. in determining the degree of acceptance of a given T, I will rely upon the Catastrophe Theory of René Thom, as described by A. Woodcock and M. Davis (1982). The simplest elementary catastrophe can be viewed as a plane convoluted in a tri-di- mensional space. The vertical axis of this space, defi ned “potential” depends on two Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ..., 29–45 ANNALES KINESIOLOGIAE • 1 • 2010 • 1 41 factors, as schematically indicated in Figure 4a. In the upper part of the fi gure the poten- tial is relatively stable, since the plane is very nearly fl at (regione stabile), so that, even large changes of one or both factors, do not lead to any substantial changes. However, when approaching the cusp, small changes of one or both factors can lead to a dramatic, “catastrophic” change of potential. After this catastrophe, a new fl at region is attained. Figure 4a: Graphic representation of the simplest catastrophe: the cusp. The “Poten- tial” (Potenziale) of a given phenomenon is assumed to depend on two control factors (Fattore di controllo). When the phenomenon is close to the cusp (piega), small chang- es of one or both control factors can lead to great “catastrophic” changes of potential. The dotted line joining A and A’ indicates such a change. The dotted area denotes an inaccessible zone (zona inaccessibile). See text for details. ANNALES KINESIOLOGIAE • 1 • 2010 • 1 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ...,29–45 42 If the potential is identifi ed with the degree of acceptance of T and the two other axes with factors presumed to foster the degree of acceptance of T, the simplest el- ementary catastrophe of Figure 2a can be applied to the philosophy of science. To this aim, I will choose two factors: 1) the agreement between facts and predictions and 2) the aesthetic appeal of T, as expressed by its “simplicity” (Figure 4b). In the period of normal science there will probably exist a leading theory (Tl) the degree of accept- ance (grado di accettazione) of which is fairly large thanks to the relatively good (alto) “explicative power” and “simplicity”. As such, Tl will fare in the lower right part of Figure 4b: The degree of acceptance of a given theory by the scientifi c community is here assumed to depend on two control factors: “simplicity” (semplicità) and “explica- tory power” (potere esplicativo). See text for details. Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ..., 29–45 ANNALES KINESIOLOGIAE • 1 • 2010 • 1 43 the plane of Figure 4b. In the revolutionary periods (as defi ned by Thomas Kuhn) new facts emerge that are not easily reconcilable with Tl which therefore moves to the left towards the zone of low (basso) explicative power, thus approaching the cusp. If this is attained, a catastrophic change of the degree of acceptance of Tl will occur, bringing it in the upper left part of the plane. To avoid the decrease of explicative power, Tl can be modifi ed by “ad hoc” constructs which, however, will detract from its simplicity. Hence the so modifi ed Tl (Tl’) will move towards the upper right part of the plane towards the zone of low (bassa) simplicity. In either case, the decrease of explicatory power or of simplicity will stimulate the scientifi c community to produce new theories or to refi ne alternative existing ones. Thanks to the emergence of new facts, these new competing theories (Tc) will be brought towards the cusp of the plane, thus moving from the upper left to the lower right part of it. If this is the case, Tc will become the new leading theory. It goes without saying that instead of two control factors (explicatory power and simplicity) as was the case here, three or more can be selected. In this case, the topo- logical representation of the catastrophe becomes more complicated, being described by an n-dimensional structure, convoluted in an (n+1)-dimensional space, where n is the number of factors assumed to control the degree of acceptance of T. Truth There are more things in Heaven and Earth, Horatio, than are dreamt of in your philosophy. 6 The aim of what follows is to discuss briefl y the meaning of scientifi c Truth within the conceptual framework summarised in the preceding sections. To this aim, I will assume that a “Theory of Everything” (TE) has been achieved and condensed in an appropriate formal system. A formal system is defi ned as a set of axioms, a set of operating rules and a set of symbols (an alphabet). A theorem is hereby defi ned as a string of symbols which can be arrived at in a fi nite number of steps from the axioms, operating on the alphabet, by means of the canonical operating rules. Thus, theorems are “True” since they are con- sistent with axioms. It also follows that within any given theory “Truth” can be defi ned as the sum total of the appropriate theorems. As shown in 1931 by Kurt Gödel (1906-1978), in a suffi ciently powerful formal system, there exists strings of symbols that are demonstrably true but that can not be arrived at in a fi nite number of steps. As such, they are not theorems. Thus the set of true 6 William Shakespeare. Hamlet I, V: 166-167. ANNALES KINESIOLOGIAE • 1 • 2010 • 1 Pietro Enrico di PRAMPERO: A BRIEF HISTORY OF HUMAN POWERED FLIGHT: FROM PHYSIOLOGY ...,29–45 44 string that can be expressed in a suffi ciently powerful formal system is larger than the set of theorems and comprehends it (Nagel & Newman, 1958; Hofstadter & Douglas, 1979). This line of reasoning shows that the theorems of TE, derived from its axioms, on the bases of its canonical rules will constitute only a subset of all true strings. The conclusion that TE is incomplete in the Gödelian sense seems therefore inescapable. CONCLUSIONS The preceding discussion was based on the correspondence theory of truth. Indeed, it was implicitly assumed that a bi-univocal correspondence does exist between our mental image of the world and the world itself. It was also assumed that the scientifi c language describing our mental image of the world is public and that, as such, it is shared by the scientifi c community at large. If we adopted the more conservative view that science is a “language game” in the Wittgensteinian sense, then any correspondence between the scientifi c description of the world and the world per se, is bound to become a meaningless expression. If this is so, we are also inevitably bound to conclude with the fi nal gnomic sentence of Wittgen- stein Tractatus Logico-Philosophicus: “Wovon man nicht, sprechen can, darüber muß man schweigen.”7 REFERENCES Feyerabend, P. K. (1975). Against Method. London: New Left Books. Grosser, M. (1991). Gossamer Odyssey. The Triumph of Human-Powered Flight. New York: Dover Publication, inc. Hacking, I., Feyerabend, P. K., Hacking, I., Kuhn, T. 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